Tightness conjecture for the torsion of random hypertrees

Let $p$ be a prime, and let $T_n$ be the random hypertree in the paper. Write $\Gamma_{n,p}$ for the $p^\infty$-torsion subgroup of $H_1(T_n,\mathbb{Z})$. A family of random variables is tight if, for every $\varepsilon>0$, there is a $K$ such that the probability of exceeding $K$ is less than $\varepsilon$ for all sufficiently large $n$.

Tightness conjecture. For every prime $p$, $\Gamma_{n,p}$ is tight:

$$\mathbb{P}(|\Gamma_{n,p}|>K)<\varepsilon$$

for every $\varepsilon>0$ and some $K$, for all sufficiently large $n$. This would also imply that $\dim H_1(T_n,\mathbb{F}_p)$ is tight.

The conjecture concerns bounded-order behavior of prime-power torsion in the first homology of random hypertrees; the source presents it as open.